Finite Group Databasebeta

← #10751705

Subgroups — #10751705

Order 768.

1706 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C2×#4947C212yesmaximal
3C2×#4951C211yesmaximal, characteristic
4C3×#1439C211yesmaximal, characteristic
5#71567C211yesmaximal, characteristic, complemented
6#71547C212yesmaximal
7#71546C211yesmaximal, characteristic
8#69424C212yesmaximal, complemented
9#69412C211yesmaximal, characteristic, complemented
10#69411C212yesmaximal, complemented
11#69403C211yesmaximal, characteristic, complemented
12#16602—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×#217C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C2×#888—28no—
25C2×#888—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4951—24no—
39#4947—28no—
40C22×C4⋊Q8—31no—
41C2×#393—32no—
42C2×#388—31no—
43#1446—32no—
44#1442—31no—
45#1442—31no—
46#1441—32no—
47#1439—31no—
48#1439—32no—
49#1439D311yescharacteristic
50#1434—31no—
51C25×C3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—42no—
65C22×Dic6—42no—
66C22×Dic6—42no—
67C22×Dic6—42no—
68C22×Dic6D418yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—42no—
75C2×C3×C22⋊C4—42no—
76C2×C3×C22⋊C4Q814yescomplemented
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—24no—
82C2×#217—42no—
83C2×#217—42no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3D414yes—
93#888—232no—
94C3×#95—22no—
95C3×#95—42no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C23×Q8—32no—
107C2×C4⋊Q8—64no—
108C2×C4⋊Q8—68no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C22⋊C4—31no—
114C22×C22⋊C4—31no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4C2×D311yescharacteristic
117C22×C22⋊C4C2×D312yes—
118C22×C42—31no—
119#393—68no—
120#388—64no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D312yes—
133C24×C3—21no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—21no—
137C24×C3—21no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3C2×Q812yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×Q812yes—
167C22×C3×C4C2×Q812yes—
168C22×C3×C4#4112yes—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3C2×D414yes—
199C22×Dic3C2×D414yes—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D412yes—
202C22×Dic3#4114yes—
203C2×Dic6—216no—
204C2×Dic6—48no—
205C2×Dic6—48no—
206C2×Dic6—48no—
207C2×Dic6—48no—
208C2×Dic6—48no—
209C2×Dic6—48no—
210C3×C4⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—48no—
213#217—48no—
214#217—44no—
215#217—44no—
216C4⋊Dic3—28no—
217C4⋊Dic3—44no—
218Dic3⋊C4—216no—
219C4×Dic3—28no—
220C25C22×D311yescharacteristic
221C22×Q8—38no—
222C22×Q8—64no—
223C22×Q8—64no—
224C22×Q8—122no—
225C22×Q8—122no—
226C22×Q8—122no—
227C22×Q8—122no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—31no—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C4⋊Q8—632no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—64no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—122no—
259C2×C22⋊C4—122no—
260C2×C22⋊C4Dic614yescomplemented
261C2×C42—34no—
262C2×C42—64no—
263#95—22no—
264#95—42no—
265#95—64no—
266#95—62no—
267#95—64no—
268#95—64no—
269#95—62no—
270#95—64no—
271#95—62no—
272#95—122no—
273#95—122no—
274#95—122no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3C22×D411yescharacteristic
299C23×C3C22×D412yes—
300C23×C3C22×Q811yescharacteristic
301C23×C3C2×#4111yescharacteristic
302C23×C3C2×#4112yes—
303C23×C3H(5,𝔽2)11yescharacteristic
304C23×C3H(5,𝔽2)12yes—
305C23×C3H(5,𝔽2)11yescharacteristic
306C23×C32−1+411yescharacteristic
307C23×C32−1+412yes—
308C23×C32−1+411yescharacteristic
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4C22⋊Q814yes—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3#12014yes—
366C2×Dic3#12414yes—
367Dic6—432no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24C2×Dic612yes—
379C24C23×D311yescharacteristic
380C2×Q8—616no—
381C2×Q8—128no—
382C2×Q8—128no—
383C2×Q8—128no—
384C2×Q8—128no—
385C2×Q8—128no—
386C2×Q8—128no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—32no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—34no—
410C22×C4—34no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4C2×Dic612yes—
441C22×C4C2×Dic612yes—
442C22×C4D3×D414yes—
443C22×C4D3×D412yes—
444C22×C4#23712yes—
445C4⋊C4—28no—
446C4⋊C4—616no—
447C4⋊C4—68no—
448C4⋊C4—124no—
449C22⋊C4—28no—
450C22⋊C4—48no—
451C22⋊C4—124no—
452C22⋊C4—124no—
453C22⋊C4—128no—
454C42—68no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3C2×#12011yescharacteristic
467C22×C3C2×C22⋊Q811yescharacteristic
468C22×C3C2×#12411yescharacteristic
469C22×C3#53412yes—
470C22×C3#53412yes—
471C22×C3#53511yescharacteristic
472C22×C3#53511yescharacteristic
473C22×C3#53611yescharacteristic
474C22×C3#53611yescharacteristic
475C22×C3#54211yescharacteristic
476C22×C3#54212yes—
477C22×C3#54212yes—
478C22×C3#54211yescharacteristic
479C22×C3#54411yescharacteristic
480C22×C3#54411yescharacteristic
481C22×C3#54812yes—
482C22×C3#54812yes—
483C22×C3#54812yes—
484C22×C3D4×Q812yes—
485C22×C3#55412yes—
486C22×C3#55512yes—
487C22×C3#56312yes—
488C22×C3#56312yes—
489C3×C4—24no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—82no—
493Dic3—24no—
494Dic3—24no—
495Dic3—44no—
496Dic3—81no—
497Dic3—81no—
498Dic3—82no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23C22×Dic611yescharacteristic
523C23C2×D3×D412yes—
524C23C2×D3×D411yescharacteristic
525C23C2×#23711yescharacteristic
526C23#100412yes—
527C23#100411yescharacteristic
528C23#100711yescharacteristic
529C23D3×#4112yes—
530C23#100911yescharacteristic
531C23#101012yes—
532C23#101011yescharacteristic
533Q8—1232no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—34no—
544C2×C4—34no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—121no—
579C2×C4—122no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—121no—
589C2×C4—122no—
590C2×C4—121no—
591C2×C4#87814yes—
592C2×C3—82no—
593C2×C3#240911yescharacteristic
594C2×C3#243611yescharacteristic
595C2×C3#256311yescharacteristic
596C2×C3#261311yescharacteristic
597C2×C3#262811yescharacteristic
598C2×C3#264011yescharacteristic
599C2×C3#272912yes—
600C2×C3#272912yes—
601C2×C3#273112yes—
602C2×C3#273112yes—
603C2×C3#282111yescharacteristic
604C22—44no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—82no—
615C22C2×#87811yescharacteristic
616C22#550011yescharacteristic
617C22#551911yescharacteristic
618C22D4×Dic612yes—
619C22#553912yes—
620C22D3×#12011yescharacteristic
621C22#558912yes—
622C22#559812yes—
623C22#560311yescharacteristic
624C22#560512yes—
625C22#560711yescharacteristic
626C22#561612yes—
627C22#562811yescharacteristic
628C22#563312yes—
629C22#564311yescharacteristic
630C22#564612yes—
631C22#564811yescharacteristic
632C22#565112yes—
633C22#566212yes—
634C22D3×#12411yescharacteristic
635C22#567712yes—
636C22#571012yes—
637C22#571511yescharacteristic
638C4—24no—
639C4—42no—
640C4—44no—
641C4—64no—
642C4—64no—
643C4—82no—
644C4—124no—
645C4—242no—
646C4—241no—
647C4—241no—
648C3#1660211yescharacteristic, complemented, Sylow
649C2—82no—
650C2#7395411yescharacteristic
651C2#7454012yes—
652C2#7468911yescharacteristic
653C2#7570712yes—
654C2#7636111yescharacteristic
655C2#7676011yescharacteristic
656C2#7678111yescharacteristic
657C2#7704612yes—
658C2#7737112yes—
659C2#7754711yescharacteristic
660C2#7809211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1439C2
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#16602
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#888C2semidirect
5. #71567C2semidirect
8. #69424C2semidirect
9. #69412C2semidirect
10. #69411C2semidirect
11. #69403C2semidirect
76. C2×C3×C22⋊C4Q8semidirect
260. C2×C22⋊C4Dic6semidirect
648. C3#16602semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#8882
C2×#49472
C2×#49512
C3×#14392
#715672
#715472
#715462
#694242
#694122
#694112
#694032
#166023